cohomology

complex geometry

# Contents

## Idea

### Pure Hodge structure

A Hodge structure (or pure Hodge structure, for emphasis) is a (bi-)grading structure on cohomology groups – called a Hodge decomposition – of the kind that is exhibited by the de Rham cohomology/complex-ordinary cohomology of compact Kähler manifolds, according to the Hodge theorem. A Hodge structure is said to be of weight $d$ if it behaves like the cohomology of a Kähler manifold of dimension $d$.

If instead of considering a single cohomology group one considers the cohomology groups of a parameterized collection of spaces – hence the cohomology sheaves/stacks – then one speaks of variation of Hodge structure (of a given weight).

By a central theorem of Hodge theory (recalled as theorem 1 below) the traditional (and original) filtration on the complex cohomology of a Kähler manifold induced by the harmonic differential forms generalizes to a filtration of the complex-valued ordinary cohomology of any complex analytic space which is simply given by the canonical degree-filtration of the holomorphic de Rham complex.

This means that ordinary differential cohomology in the guise of Deligne cohomology is nothing but the homotopy pullback of a stage of the Hodge filtration along the “ Chern character ” map from integral to complex cohomology. (A point of view highlighted for instance in Peters-Steenbrink 08, section 7.2). Viewed this way Hodge structures are filtrations of stages of differential form cycle refinements of Chern characters that appear in the general definition/characterization of differential cohomology, as discussed at differential cohomology hexagon starting around the section de Rham coefficients

This modern point of view is also crucial for instance in the characterization of an intermediate Jacobian (see there) as the subgroup of Deligne cohomology that is in the kernel of the map to Hodge-filtering stage of ordinary cohomology. See at intermediate Jacobian – characterization as Hodge-trivial Deligne cohomology.

### Mixed Hodge structure

A mixed Hodge structure is a filtration on cohomology groups – called a Hodge filtration – such that the associated graded object has pure Hodge structure of weight $k$ in each degree $k$. The archetypical example exhibiting this is the cohomology of complex varieties that have singularities (Deligne 71 Deligne 74).

## Definition

Historically, Hodge structures originate in the special structure induced on the de Rham cohomology groups of a compact Kähler manifold by the existence of harmonic differential forms. Below we first discuss this canonical Hodge structure

But it turns out that this Hodge structure only depends on the natural degree-filtration on the holomorphic de Rham complex and hence more generally there is canonical Hodge structure

Abstracting from here one defines Hodges structures

### On the cohomology of a Kähler manifold

Let $X$ be a compact Kähler manifold and write $H^{p,q}(X)$ for its space of harmonic differential forms, equivalently, via the Hodge isomorphism, its Dolbeault cohomology in bidegree $(p,q)$.

Notice that by the de Rham theorem there are canonical maps

$H^{p,q}(X)\to H^{p+q}(X,\mathbb{C})$

to ordinary cohomology of $X$ with complex coefficients.

###### Definition

The Hodge filtration on the cohomology of $X$ is the filtered complex structure given by the direct sum

$F^p H^k(X, \mathbb{C}) \coloneqq \underset{k-q \geq p}{\oplus} H^{k-q,q}(X) \,.$
###### Example

The full Hodge filtration of degree-2 cohomology is

\begin{aligned} F^0 H^2(X,\mathbb{C}) & = H^{0,2}(X) \oplus H^{1,1}(X) \oplus H^{2,0}(X) \\ F^1 H^2(X,\mathbb{C}) & = \;\;\;\;\;\;\; H^{1,1}(X) \oplus H^{2,0}(X) \\ F^2 H^2(X,\mathbb{C}) & = \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; H^{2,0}(X) \end{aligned}
###### Example

For all $p$ the mid-dimensional Hodge filtration stage in even total degree is

$F^p H^{2p} = H^{p,p}(X) \oplus H^{p+1,p-1}(X) \oplus \cdots \oplus H^{2p,0}(X) \,.$

### On the cohomology of a complex analytic space

###### Definition

For $X$ a complex analytic space, write

$\Omega^\bullet_X \coloneqq (\mathcal{O}_X \stackrel{\partial}{\longrightarrow} \Omega^1_X \stackrel{\partial}{\longrightarrow} \Omega^2(X) \stackrel{\partial}{\longrightarrow} \cdots)$

for its holomorphic de Rham complex.

###### Remark

Notice the relation to complex cohomology given by

$H^k(X,\mathbb{C}) \simeq H^k(X,\Omega^\bullet_X) \,.$
###### Remark

The holomorphic de Rham complex is naturally filtered by degree with the $p$th filtering stage being

$F^p \Omega^\bullet_X \coloneqq (0 \to \cdots \to \Omega^p_X \stackrel{\partial}{\longrightarrow} \Omega^{p+1} \stackrel{\partial}{\longrightarrow} \cdots) \,.$

Notice that here $\Omega^p$ is still regarded as sitting in degree $-p$, one just replaces by 0 in the holomorphic de Rham complex the groups of differential forms of degree less than $p$.

###### Definition

The Hodge filtration on $H^\bullet(X,\mathbb{C})$ is defined to be the filtration with $p$th stage the image of the hyper-abelian sheaf cohomology with coefficients in the $p$th filtering stage of the holomorphic de Rham complex inside that with coefficients the full de Rham complex:

$F^p H^k(X,\mathbb{C}) \coloneqq im \left( H^k(X, F^p \Omega^\bullet_X) \to H^k(X, \Omega^\bullet_X) \right)$
###### Theorem

When the compact complex manifold $X$ happens to have the structure of a Kähler manifold then the Frölicher spectral sequence degenerates at the $E_1$ page which implies that def. 3 coincides with the traditional definition via harmonic differential forms, def. 1:

$\underset{k-q \geq p}{\oplus} H^{k-q,q}(X) \simeq im \left( H^k(X, F^p \Omega^\bullet_X) \to H^k(X, \Omega^\bullet_X) \right) \,.$
###### Remark

The equivalence in theorem 1 is exhibited by the following morphism.

Write

$tot(\Omega^{\bullet \geq p, \bullet}, \mathbf{d}= \partial + \bar \partial)$

for the holomorphically truncated de Rham complex, as indicated, thought of as the total complex of the Dolbeault double complex

$\array{ \Omega^{p,0} &\stackrel{\bar \partial}{\to}& \Omega^{p-1,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \Omega^{p+1,0} &\stackrel{\bar \partial}{\to}& \Omega^{p,1} &\stackrel{\bar \partial}{\to}& \cdots \\ \downarrow^{\mathrlap{\partial }} && \downarrow^{\mathrlap{\partial }} \\ \vdots && \vdots } \,.$

Since this is in each row the Dolbeault resolution of the given sheaf of holomorphic differential forms, this total complex is indeed quasi-isomorphic to the (truncated) holomorphic de Rham complex.

The total complex is in degree $-k$ given by $\underset{k-q \geq p}{\oplus} \Omega^{k-q, q}$ and hence globally defined closed $(k-q \geq p,q)$-forms naturally inject into

$H^k(X, tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial) ) \simeq H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k]) \,.$

Therefore given a representative $\alpha \in \Omega^{p,q}_{cl}(X)$ of $[\alpha] \in H^{p,q}(X)$ it is canonically sent along

$\underset{k-q\geq p}{\oplus} \Omega^{p,q}_{cl}(X) \simeq \underset{k-q\geq p}{\oplus} H^0(X, \Omega^{p,q}_{cl}) \to H^0(X,tot(\Omega^{\bullet\geq p, \bullet}, \mathbf{d} = \partial + \bar \partial)[-k]) \,.$

This map exhibits the equivalence in theorem 1 (e.g. Voisin, section 1.1.2).

Dually,

$\Omega_{hol}^{\leq k} \simeq tot( \Omega^{\bullet \leq k, \bullet}) \,.$

This plays a role in the discussion of intermediate Jacobians, where for $dim_{\mathbb{C}}(X)= k+1$ we have

$H^{2k+1}(X,\mathbb{R}) \simeq H^{2k+1}(X,\mathbb{C}) / F^{k+1} H^{2k+1}(X,\mathbb{C}) \simeq H^{2k+1}(X, \Omega_{hol}^{\bullet \leq k}) \,.$

Here a real differential $(2k+1)$-form

$\alpha = \overline{\alpha^{0,2k+1}} + \overline{\alpha^{1, 2k}} + \cdots + \alpha^{1, 2k} + \alpha^{0,2k+1}$

injects via its pieces in

$\Omega^{p \leq k, 2k+1-p}(X) \simeq H^0(X, \Omega^{p \leq k, 2k+1-p}) \to H^0(X, tot(\Omega^{\bullet \leq k, \bullet})[-k]) \simeq H^k(\Omega_{hol}^{\bullet\leq k}) \,.$

### Generally on an abelian group

###### Definition

For $H_{\mathbb{Z}}$ an abelian group, a Hodge structure of weight $k \in \mathbb{Z}$ on $H_{\mathbb{Z}}$ is a direct sum decomposition of its complexification

$H_{\mathbb{C}}\coloneqq H_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{C}$

into complex vector spaces $H^{p,q}$ with $p +q = k$ of the form

$H_{\mathbb{C}} \simeq \underset{p+q = k}{\oplus} H^{p,q}$

such that $H^{q,p}$ is the complex conjugate of $H^{p,q}$:

$H^{p,q} = \overline{H^{q,p}} \,.$

This is an equality of the underlying sets of the complex vector spaces.

With this the above def. 1 has the following verbatim generalization

###### Definition

Given a Hodge structure $H_{\mathbb{Z}}, \{H^{p,q}\}$ of weight $k$, def. 4, then the associated Hodge filtration on $H_{\mathbb{C}}$ is the filtered complex structure given by the direct sum

$F^p H_{\mathbb{C}} \coloneqq \underset{k-q \geq p}{\oplus} H^{k-q,q} \,.$

## References

Textbook accounts include

The notion of mixed Hodge structures was introduced in

• Pierre Deligne, Théorie de Hodge II, Publ. Math. I.H.E.S, 40, 5–58 (1971)

• Pierre Deligne, Théorie de Hodge III, Publ. Math., I. H. E. S, 44, 5-77 (1974)

A review is in section 8.4 of